The “chromatic/parallel” keyboard has 12 one-way “up and down” keys with 3 tangents each one.

Tuning: V, IV or IV, V.

The 36 semitones obtainable from each string occur in 1 arrangement coinciding with the tuning, they are available 3 at a time in 12 single choices which cannot be combined with each other. Therefore, the virtual range of 2 octaves minus 1 semitone is reduced to its half.

Francisco Luengo, who built this kind of keyboard in the eighties writes: “The keys are eleven … twelve available sounds, surely a chromatic octave. This fact doesn’t imply that organistrum was intended to play other than modal music, but, certainly, it was an instrument for transposition, able both to change the pitch of any composition and to carry out all exachords combinations” (Francisco Luengo in: El Portico de la Gloria. Musica, Arte y pensamento. “Cuadernos de Musica en Compostela II” Santiago de Compostela, 1988 , p.111). Approximately the same words in Christian Rault, La reconstitution de l’Organistrum (available on Google).

My only observation concerns the impossibility of transposing an 8 sounds average gregorian melody entirely within a single octave.

The “diatonic/polyphonic” type has 10 one-way “up and down” keys and 1 both “up and down” and 180° spinning key.

Tuning: I, VIII.

3 keys operate on the bass string, 4 (3+1) on the middle one, 5 on the treble. 1 key of the latter group can be spun 180 degrees and operated both on the middle and on the treble strings alternately, producing different sounds. Then 14 sounds are available and they can be combined operating the independent keys, 2 at a time, as follows:

(4x5)+ (4x6) + (5x6) = 20+24+30 = 74

But, since part of these 74 combinations generates 22 dissonant intervals either rarely or never performed (II and VII) plus 4 tritones, the total number of actual combinations amounts to 74 - 22 - 4 = 48 .

On this diatonic keyboard, with this tuning: A,A, a (whole scale: one octave plus one fifth) it is possible to perform music in 2 modes: Protus plagalis and Protus autenticus. Some advantages are that the three “mother-strings” Do, Re and Mi lie on the bass string and the main exachords: naturalis, durus and mollis are all represented, while f# (ficta) introduces an additional “false” exachord.

In 12th century two voices polyphonic compositions, whatever the Mode, a range of 20 to 30 combinations of sounds is requested.

The amount of 48 is enough to serve no more than 1 authentic mode and its plagal, considering that they have a good deal of sounds in common.

In a mathematical way:

48:2 = 24

24<30

3.The “chromatic/polyphonic” keyboard has 11 both “up and down” and 360° spinning keys bearing 5 tangents on 4 different positions each one,

Tuning: IV,V or V,IV

The 12 semitones obtainable from each string can be managed separately by using 1 key at a time. To these 36 choices some others have to be added: the 12 “organum parallelum” choices on treble and middle strings keeping the bass as a drone. Thus 36+12 = 48 choices in total.

Furthermore, by managing 2 keys at a time, 6 combinations of sounds are available for each couple. Since keys combinations are 12x12 = 144 in total, then the sum of all possible combinations of sounds amounts to 144x6 = 864,

to which the first group of 48 sounds has to be added: 912 possible combinations of sounds in total.

12x12 = 144

144x6 = 864

864+48 = 912

But, since in all 12th century music compositions, no matter which modal transposition occurs, no more than 8 keys are required, performing combinations are, as follows:

8x8 = 64

64x6 = 384

384+48 = 432

Then, all combinations that give dissonant intervals rarely or never performed (II and VII) have to be subtracted from this number: 44 dissonances between the outer strings and 4 for each semitone of the adjacent strings:

44+( 4x11) + (4x11) = 132 in sum

Adding to this amount 28 tritones and 11 fourths (or fifths, depending on middle string pitch) not to be performed by the same key on the bass and middle strings: 132+28+11 = 171 combinations to be avoided.

Finally: 432-171 = 261 useful combinations.

In 12th century two voices polyphonic compositions, whatever the Mode, an average amount of 20 to 30 combinations of sounds is requested.

Sure enough, this advanced keyboard, actually extended over 2 octaves minus a semitone, allows us to play in each of the 8 modes.